How do I shift a vertex to establish a single, flat surface on a quad?
If you only care about planar faces in general, then there's an automatic way for that by selecting Mesh > Clean Up > Make Planar Faces.

If you really want to only move a specific vertex of the quad, you can


move the 3D Cursor to one of the three other vertices (Select the vertex, Shift+S > Cursor to Selected) and switch to Pivot Point > 3D Cursor (. key)
Shift-select all three vertices that should define your plane and should not move
Hit Ctrl+Alt+Space to define a custom Transform Orientation
Select the vertex that you want to move and scale it locally along the Z-axis of the new transform orientation: SZZ 0 Enter